Proving $\bigcup \mathcal{P}(\mathcal{A})=\mathcal{A}$ I'm working on operations on collections of sets and I've run aground. I'm trying to prove that if $\mathcal{A}$ is a collection of sets $\mathcal{A}_i, i=1,2,...$, then $$\bigcup{\mathcal{P}(\mathcal{A})}=\mathcal{A}$$ But how do I write the notation for the power set of a collection of sets? They are sets of sets, so $\\{\mathcal{A}, \\{\mathcal{A}_1\\},...\\{\mathcal{A}_1,\mathcal{A}_2\\},...\\}$... But how do I write that notationally?

1. For any $a\in A$, we have $a\in \\{a\\}\subseteq A$; hence $a\in \bigcup \mathcal{P}(A)$.

2. For any $a\in \bigcup \mathcal{P}(A)$, there is some $B\in \mathcal{P}(A)$ with $a\in B$. But then $a\in B\subseteq A$, so $a\in A$.




The nature of the elements of $A$ are irrelevant; in your case they happen to be sets but it doesn't matter.